Properties

Label 30.0.132...687.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.321\times 10^{46}$
Root discriminant $34.46$
Ramified primes $7, 11$
Class number $80$ (GRH)
Class group $[2, 2, 2, 10]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 5*x^28 - 6*x^27 + 20*x^26 - 27*x^25 + 75*x^24 - 46*x^23 + 211*x^22 - 109*x^21 + 617*x^20 - 357*x^19 + 1911*x^18 - 1372*x^17 + 2909*x^16 - 2210*x^15 + 4354*x^14 - 2262*x^13 + 5693*x^12 + 2042*x^11 + 3092*x^10 + 1302*x^9 + 2375*x^8 + 290*x^7 + 1798*x^6 - 511*x^5 + 146*x^4 - 41*x^3 + 12*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^30 - x^29 + 5*x^28 - 6*x^27 + 20*x^26 - 27*x^25 + 75*x^24 - 46*x^23 + 211*x^22 - 109*x^21 + 617*x^20 - 357*x^19 + 1911*x^18 - 1372*x^17 + 2909*x^16 - 2210*x^15 + 4354*x^14 - 2262*x^13 + 5693*x^12 + 2042*x^11 + 3092*x^10 + 1302*x^9 + 2375*x^8 + 290*x^7 + 1798*x^6 - 511*x^5 + 146*x^4 - 41*x^3 + 12*x^2 - 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 12, -41, 146, -511, 1798, 290, 2375, 1302, 3092, 2042, 5693, -2262, 4354, -2210, 2909, -1372, 1911, -357, 617, -109, 211, -46, 75, -27, 20, -6, 5, -1, 1]);
 

\( x^{30} - x^{29} + 5 x^{28} - 6 x^{27} + 20 x^{26} - 27 x^{25} + 75 x^{24} - 46 x^{23} + 211 x^{22} - 109 x^{21} + 617 x^{20} - 357 x^{19} + 1911 x^{18} - 1372 x^{17} + 2909 x^{16} - 2210 x^{15} + 4354 x^{14} - 2262 x^{13} + 5693 x^{12} + 2042 x^{11} + 3092 x^{10} + 1302 x^{9} + 2375 x^{8} + 290 x^{7} + 1798 x^{6} - 511 x^{5} + 146 x^{4} - 41 x^{3} + 12 x^{2} - 3 x + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-13209167403604364499542354001933559191813355687\)\(\medspace = -\,7^{25}\cdot 11^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.46$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(77=7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{77}(64,·)$, $\chi_{77}(1,·)$, $\chi_{77}(3,·)$, $\chi_{77}(4,·)$, $\chi_{77}(5,·)$, $\chi_{77}(71,·)$, $\chi_{77}(9,·)$, $\chi_{77}(75,·)$, $\chi_{77}(12,·)$, $\chi_{77}(15,·)$, $\chi_{77}(16,·)$, $\chi_{77}(67,·)$, $\chi_{77}(20,·)$, $\chi_{77}(23,·)$, $\chi_{77}(25,·)$, $\chi_{77}(26,·)$, $\chi_{77}(27,·)$, $\chi_{77}(69,·)$, $\chi_{77}(31,·)$, $\chi_{77}(34,·)$, $\chi_{77}(36,·)$, $\chi_{77}(37,·)$, $\chi_{77}(38,·)$, $\chi_{77}(45,·)$, $\chi_{77}(47,·)$, $\chi_{77}(48,·)$, $\chi_{77}(53,·)$, $\chi_{77}(58,·)$, $\chi_{77}(59,·)$, $\chi_{77}(60,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{43} a^{19} - \frac{17}{43} a^{18} - \frac{14}{43} a^{17} + \frac{14}{43} a^{16} + \frac{5}{43} a^{15} + \frac{16}{43} a^{14} + \frac{19}{43} a^{13} - \frac{4}{43} a^{12} - \frac{13}{43} a^{11} + \frac{14}{43} a^{10} + \frac{3}{43} a^{9} + \frac{7}{43} a^{8} + \frac{4}{43} a^{7} + \frac{4}{43} a^{6} + \frac{9}{43} a^{5} + \frac{11}{43} a^{4} + \frac{10}{43} a^{3} - \frac{20}{43} a^{2} + \frac{19}{43} a + \frac{18}{43}$, $\frac{1}{43} a^{20} - \frac{2}{43} a^{18} - \frac{9}{43} a^{17} - \frac{15}{43} a^{16} + \frac{15}{43} a^{15} - \frac{10}{43} a^{14} + \frac{18}{43} a^{13} + \frac{5}{43} a^{12} + \frac{8}{43} a^{11} - \frac{17}{43} a^{10} + \frac{15}{43} a^{9} - \frac{6}{43} a^{8} - \frac{14}{43} a^{7} - \frac{9}{43} a^{6} - \frac{8}{43} a^{5} - \frac{18}{43} a^{4} + \frac{21}{43} a^{3} - \frac{20}{43} a^{2} - \frac{3}{43} a + \frac{5}{43}$, $\frac{1}{43} a^{21} + \frac{7}{43} a^{14} - \frac{1}{43} a^{7} - \frac{7}{43}$, $\frac{1}{43} a^{22} + \frac{7}{43} a^{15} - \frac{1}{43} a^{8} - \frac{7}{43} a$, $\frac{1}{43} a^{23} + \frac{7}{43} a^{16} - \frac{1}{43} a^{9} - \frac{7}{43} a^{2}$, $\frac{1}{43} a^{24} + \frac{7}{43} a^{17} - \frac{1}{43} a^{10} - \frac{7}{43} a^{3}$, $\frac{1}{396586807} a^{25} - \frac{4350714}{396586807} a^{24} - \frac{3410001}{396586807} a^{23} + \frac{4453046}{396586807} a^{22} - \frac{3476348}{396586807} a^{21} - \frac{3558177}{396586807} a^{20} - \frac{853258}{396586807} a^{19} - \frac{158131282}{396586807} a^{18} + \frac{38712637}{396586807} a^{17} + \frac{135694806}{396586807} a^{16} + \frac{76587940}{396586807} a^{15} + \frac{61497807}{396586807} a^{14} + \frac{103663006}{396586807} a^{13} + \frac{193760508}{396586807} a^{12} + \frac{104240870}{396586807} a^{11} + \frac{143709208}{396586807} a^{10} - \frac{78603003}{396586807} a^{9} - \frac{133814790}{396586807} a^{8} + \frac{112965244}{396586807} a^{7} + \frac{194614168}{396586807} a^{6} - \frac{33651863}{396586807} a^{5} + \frac{180382075}{396586807} a^{4} - \frac{3532178}{396586807} a^{3} + \frac{45653903}{396586807} a^{2} - \frac{138003062}{396586807} a + \frac{5432656}{396586807}$, $\frac{1}{17053232701} a^{26} - \frac{17}{17053232701} a^{25} - \frac{97613540}{17053232701} a^{24} + \frac{94533106}{17053232701} a^{23} - \frac{88400513}{17053232701} a^{22} + \frac{173692367}{17053232701} a^{21} + \frac{49145536}{17053232701} a^{20} + \frac{60840015}{17053232701} a^{19} - \frac{2587182035}{17053232701} a^{18} + \frac{186363464}{17053232701} a^{17} - \frac{5718119947}{17053232701} a^{16} + \frac{1246469766}{17053232701} a^{15} + \frac{1194079629}{17053232701} a^{14} + \frac{5718763938}{17053232701} a^{13} - \frac{4030494251}{17053232701} a^{12} - \frac{842854905}{17053232701} a^{11} - \frac{4665241032}{17053232701} a^{10} - \frac{3217753510}{17053232701} a^{9} + \frac{5057589376}{17053232701} a^{8} - \frac{8312034845}{17053232701} a^{7} - \frac{378755582}{17053232701} a^{6} + \frac{7078043486}{17053232701} a^{5} - \frac{2591698761}{17053232701} a^{4} + \frac{2626265490}{17053232701} a^{3} + \frac{3009221732}{17053232701} a^{2} - \frac{6857505879}{17053232701} a - \frac{6636223328}{17053232701}$, $\frac{1}{17053232701} a^{27} - \frac{2}{17053232701} a^{25} - \frac{80954690}{17053232701} a^{24} - \frac{76517526}{17053232701} a^{23} + \frac{149285566}{17053232701} a^{22} + \frac{94953880}{17053232701} a^{21} - \frac{14683529}{17053232701} a^{20} - \frac{170494006}{17053232701} a^{19} + \frac{430275151}{17053232701} a^{18} + \frac{4744523170}{17053232701} a^{17} + \frac{503790404}{17053232701} a^{16} + \frac{6014168224}{17053232701} a^{15} - \frac{2386785948}{17053232701} a^{14} + \frac{3772655943}{17053232701} a^{13} + \frac{449498506}{17053232701} a^{12} - \frac{7494769887}{17053232701} a^{11} - \frac{4904306360}{17053232701} a^{10} - \frac{5192960232}{17053232701} a^{9} - \frac{8319847735}{17053232701} a^{8} - \frac{5923274732}{17053232701} a^{7} + \frac{36439945}{86564633} a^{6} - \frac{6648973526}{17053232701} a^{5} + \frac{2459203371}{17053232701} a^{4} + \frac{2033338395}{17053232701} a^{3} + \frac{4534251734}{17053232701} a^{2} + \frac{3915623022}{17053232701} a - \frac{3012685487}{17053232701}$, $\frac{1}{17053232701} a^{28} - \frac{133596600}{17053232701} a^{21} - \frac{6742221803}{17053232701} a^{14} - \frac{6597075107}{17053232701} a^{7} + \frac{7611643532}{17053232701}$, $\frac{1}{17053232701} a^{29} - \frac{133596600}{17053232701} a^{22} - \frac{6742221803}{17053232701} a^{15} - \frac{6597075107}{17053232701} a^{8} + \frac{7611643532}{17053232701} a$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}$, which has order $80$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1163304351}{17053232701} a^{29} - \frac{2326608702}{17053232701} a^{28} + \frac{6592057989}{17053232701} a^{27} - \frac{12408579744}{17053232701} a^{26} + \frac{28308339807}{17053232701} a^{25} - \frac{52348695795}{17053232701} a^{24} + \frac{110901681462}{17053232701} a^{23} - \frac{130290087312}{17053232701} a^{22} + \frac{269886609432}{17053232701} a^{21} - \frac{354420058938}{17053232701} a^{20} + \frac{762739886139}{17053232701} a^{19} - \frac{1090709287228}{17053232701} a^{18} + \frac{2399121339879}{17053232701} a^{17} - \frac{3680694966564}{17053232701} a^{16} + \frac{4239081055044}{17053232701} a^{15} - \frac{5422937116245}{17053232701} a^{14} + \frac{6507912307611}{17053232701} a^{13} - \frac{6839454047646}{17053232701} a^{12} + \frac{7561641206675}{17053232701} a^{11} - \frac{3370092704847}{17053232701} a^{10} - \frac{986094321531}{17053232701} a^{9} - \frac{2874137283204}{17053232701} a^{8} + \frac{49246550859}{17053232701} a^{7} - \frac{2930363660169}{17053232701} a^{6} + \frac{833313683433}{17053232701} a^{5} - \frac{2834616310174}{17053232701} a^{4} + \frac{67083884241}{17053232701} a^{3} - \frac{19388405850}{17053232701} a^{2} + \frac{5040985521}{17053232701} a - \frac{1551072468}{17053232701} \) (order $14$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 4697581952.048968 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 4697581952.048968 \cdot 80}{14\sqrt{13209167403604364499542354001933559191813355687}}\approx 0.219329283542475$ (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{7})\), 10.0.3602729712967.1, 15.15.886528337182930278529.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ $30$ $30$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ $30$ $30$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ $30$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{30}$ $30$ $15^{2}$ $30$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$